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arXiv:2408.10832 · Letters in Mathematical Physics

Boundary symmetries of (2+1)D topological orders

Kylan Schatz

Published Jun 12, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract For a G-crossed braided extension of a unitary modular tensor category $$\mathcal {C}$$ C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object $$A \in \mathcal {C}$$ A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.

DOI
10.1007/s11005-025-01953-w

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Abstract For a G-crossed braided extension of a unitary modular tensor category C\mathcal {C}C C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object A∈CA \in \mathcal {C}A∈C A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.
We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category C\mathcal{C}C, an indecomposable semisimple right C\mathcal{C}C-module category M\mathcal{M}M, a Q-system Q∈CQ\in\mathcal{C}Q∈C specifying the bulk phase, and a right QQQ-module K∈MQK\in\mathcal{M}_{Q}K∈MQ​, regarded as an object of MQop\mathcal{M}_{Q}^{\mathrm{op}}MQop​, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor MQop→BCond\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}MQop​→BCond is an equivalence, so simple boundary conditions are classified by simple objects of MQ\mathcal{M}_{Q}MQ​ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to (CM∨)rev(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}(CM∨​)rev, and the canonical action of the bulk DHR category on it agrees with the categorical action of Z1(Crev)Z_1(\mathcal{C}^{\mathrm{rev}})Z1​(Crev). Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of (CM∨)rev(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}(CM∨​)rev on MQop\mathcal{M}_{Q}^{\mathrm{op}}MQop​. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.
We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of GGG-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.
We investigate symmetry-preserving gapped boundary of (2+1)(2+1)(2+1)-dimensional [(2+1)D][(2+1)\mathrm{D}][(2+1)D] topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)f{}^{f}f-symmetric (2+1)D(2+1)\mathrm{D}(2+1)D fermionic topological phases, which are beyond chiral central charge ${c}_{\ensuremath{-}}$ and electric Hall conductivity ${\ensuremath{\sigma}}_{H}$. These obstructions are given by a simple Gauss-Milgram-type formula valid for supermodular category and regarded as a higher version of ${c}_{\ensuremath{-}}$ and ${\ensuremath{\sigma}}_{H}$.
In this work, we give a mathematical description of a chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk rational conformal field theories naturally emerge on certain gapless walls between two trivial phases.
Symmetry-protected topological (SPT) states have boundary 't Hooft anomalies that obstruct the effective boundary theory realized in its own dimension with UV completion and with an on-site G-symmetry.In this work, yet we show that a certain anomalous non-on-site G-symmetry along the boundary becomes on-site when viewed as an extended H-symmetry, via a suitable group extension 1 → K → H → G → 1. Namely, a nonperturbative global (gauge or gravitational) anomaly in G becomes anomaly free in H.This guides us to construct an exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries applicable to any SPT state of any finite symmetry group, including on-site unitary and antiunitary time-reversal symmetries.The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary in any spacetime dimension or, more naturally, by a topological emergent K-gauge theory with a global symmetry G on a 3 þ 1D bulk or above.The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry G, described by representations of H. (Applying our approach to a 1 þ 1D boundary of 2 þ 1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order.The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.)In contrast to known gapped boundaries or interfaces obtained via symmetry breaking (either global symmetry breaking or the Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension.More generally, applying our approach to SPT states, topologically ordered gauge theories, and symmetry enriched topologically ordered (SET) states leads to generic boundaries or interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.
We review the condensation completion of a modular tensor category C\mathcal{C}C, which yields a fusion 2-category ΣCΣ\mathcal{C}ΣC of separable algebras, bimodules over algebras and bimodule maps in C\mathcal{C}C. Physically, ΣCΣ\mathcal{C}ΣC is the fusion 2-category of codimension-1 defects, codimension-2 defects and instantons in the 2+12+12+1D topological order C\mathcal{C}C. We realize the rough-rough wall and eee-mmm exchange wall in Toric Code model on the lattice by deforming the Hamiltonian based on the corresponding algebraic data. We apply condensation completion to Toric Code, 3F3\mathbf{F}3F, two-laryer semion and Z4\mathbb{Z}_4Z4​ topological orders, and explicitly enumerate their 111d and 000d defects along with fusion rules. We also mention other applications of condensation completion: alternative interpretations of condensation completion of a braided fusion category; condensation completion of the category of symmetry charges and its correspondence to gapped phases with symmetry; for a topological order C\mathcal{C}C, one can find all gapped boundaries of the stacking of C\mathcal{C}C with its time-reversal conjugate through computing the condensation completion of C\mathcal{C}C.